The spectrum of prime congruences of a pair

Continuing the study of the structure of semirings, we turn to the spectrum of prime congruences. Joo and Mincheva developed an elegant theory in the special case of idempotent semirings, which is generalized here to semiring pairs,'' which include supertropical semirings and various classes of hyperrings. Our main result is that the Joo-Mincheva spectrum can be embedded into the prime spectrum of a semiring pair, and the mapping is onto when the pair satisfies a mild restriction which we call positive $e$-type.‘’